Whew!
OK... a woman alone, using normal common sense should be fine in Panama. I know many single women happily living here. Panamanian men think gringo women are really interesting though, though they have never acted disrespectful in my experience.
Panamanians are generally scared to death of dogs, and I'm sure they wouldn't come near a pit bull. But, they highly respect your decisions and won't think less of you for having scary dogs. They won't come in your yard though, which may not be such a bad thing... woman living alone, etc.
https://www.youtube.com/watch?v=Rh_ClPYNzEw This comes to mind about finding housing. Maybe you can connect with other expats in your area and get recommendations.
http://kraemerlaw.com/en/ a very good lawyer
Groceries - I am not very familiar with your area but you are close to Coronado, land of many gringos so I imagine there will be supermarkets catering to their tastes. I think there are also activities in Coronado where you can connect with other expats and ask them questions. Expats are generally very happy to help a newcomer get settled.
Internet, depending on where you are is usually fine, not lightning fast but good enough for about any need.
You will probably find it easy to make friends. As I said, there are many expats nearby and if you speak Spanish, you can't possibly be lonely. Panamanians are very friendly, welcoming, and happy to become your new family.
Bring as little as possible - electronics (computer, tablet, phone, camera etc) as those things are at least as expensive here. Of course passport, important papers, more than one credit card (you don't want to be without access to money) Otherwise, it's hard to know what you will need here until you are living your new life for a while. I came with one suitcase and didn't even use everything in that. But, on this there are many differing opinions, so it's an individual decision.
You can use your phone, but I suggest you get it unlocked by your provider before you leave. Then, you only need a new sim card here (and whatever plan will work for you)
You can probably get by with a scooter or bicycle. There are a lot of inexpensive buses and taxis. If you find this isn't working, you can look for a used car, but used cars are surprisingly expensive because of the taxes and fees and whatever they tack on. Or, maybe you can get lucky with some expat who is leaving and needs to sell their car.
Medical care is inexpensive, so you can pay out of pocket but if you land in the hospital, they generally want a deposit and credit card number so they will be sure they are paid. If you feel insurance is the way to go, most people get an international policy. I hear that is cheaper if it doesn't include USA care.
Yes, you can shop on line, but you'd probably want a mail service like Mailboxes Etc. Again, a good question for other expats living in your area.